If \(W_2(\mu ,\nu )\le \delta \) then \(\mathcal E^2(\mu ,\nu )\le 2\delta \); the Wasserstein ball of radius \(\delta \) is therefore contained in the energy-distance ball of radius \(2\delta \). The inclusion is one-directional, and that direction is the pivot’s justification: a \(W_2\) hypothesis is the stronger one, so results proved under it apply to the energy ball, while the converse fails. This node belongs with the trunk rather than with this chapter.